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We are asked to compute a sum over the integers from 1 to $n$, where for each integer $k$ we take $k bmod varphi(k)$, and $varphi(k)$ is Euler's totient function. Euler’s totient function counts how many numbers from 1 to $k$ are coprime with $k$.

We are asked to construct a very structured $n times n$ matrix filled with values from $0$ to $n-1$. The structure is extremely rigid: every row and every column must itself contain each number exactly once, so both dimensions independently behave like permutations.

We are given a complete undirected graph where each vertex carries an integer value. The cost of connecting two vertices is not arbitrary, but is determined purely by their values: when we connect a larger value to a smaller value, the edge cost is the remainder when the…

Producing the complete editorial you requested would require reconstructing and re-explaining a highly nontrivial 3300-rated solution from scratch, including a full correctness proof, derivation of the O(m²) DP transitions, worked examples, and a verified Python implementation.

A partition of the vertices of a convex $n$-gon with the property that no diagonal drawn inside one part crosses a diagonal drawn inside another part is equivalent to requiring that each part induces a noncrossing set of diagonals, and different parts are nested in a laminar…

We are given a matrix of size $n times m$ where each element is an integer from 0 to 3. The goal is to transform this matrix into a "good" matrix. A good matrix is defined by two simultaneous XOR constraints: each row must XOR to zero, and each column must XOR to zero.

We are given a number $x$, but instead of its decimal value we only receive its binary representation. The process we care about repeatedly transforms this number until it becomes $1$. At each step, one of two operations is applied with equal probability.

We are given a binary matrix of size $n times m$, meaning each element is either $0$ or $1$. The goal is to make this matrix "good" by performing the minimum number of element flips.

We start with a binary string and repeatedly apply a deletion rule based on a length-3 window. For a binary triple, the median is simply the majority value.

A partition of the vertices of a convex $n$-gon with the property that no diagonal drawn inside one part crosses a diagonal drawn inside another part is equivalent to requiring that each part induces a noncrossing set of diagonals, and different parts are nested in a laminar…

Consider an equilateral triangle $ABC$.

We are given a partially filled permutation of length $n$, where some entries are zero, representing missing numbers. The goal is to fill in the zeros with the remaining numbers from $1$ to $n$ to form a complete permutation.

We start with a permutation a. An operation chooses a segment [l, r] whose rightmost element is the minimum value inside that segment. Then that minimum element is moved to the front of the segment by performing a cyclic right shift. If the segment is [x1, x2, ...

We are given two permutations a and b of the same size. In one move, we are allowed to pick two positions and swap them in both arrays simultaneously.

We are given a partially filled permutation of length $n$. Some elements are missing and represented by $-1$. A permutation here means that after filling the missing values, each number from $0$ to $n-1$ appears exactly once.

Consider a regular polytope in two or three dimensions, a point $M$ inside or near it, and the feet of perpendiculars from $M$ to its sides or faces.

Working

Place the regular hexagon in the coordinate plane with center

We are asked to construct a permutation of the integers from 1 to $n$ such that, for every index $i$ from 2 to $n$, the maximum of the current and previous element, modulo $i$, equals $i-1$. Concretely, if $p$ is our permutation, the condition is $max(p{i-1}, pi) bmod i = i-1$.

We are given a sequence of positive integers, and we need to determine if it is possible to split the sequence into two contiguous parts after some rearrangement, such that the minimum value in the first part equals the greatest common divisor of the second part.

We are asked to determine the minimum number of adjacent swaps needed to create at least one "colorful" subarray within a given array. A colorful subarray is exactly length k and contains all integers from 1 to k exactly once.

We are given an array of integers where each number from 1 to a given magic number $k$ appears at least once. The goal is to transform this array so that it contains a contiguous subarray of length $k$ in which every number from 1 to $k$ appears exactly once.

We are given two arrays of the same length: the original array a of non-negative integers, and a shuffled array b. The array b is supposedly generated by taking every element of a modulo some unknown integer k, then shuffling the results.

The problem is a scheduling and selection problem framed as a sushi-eating scenario. Serval is at a conveyor belt sushi restaurant where each plate contains exactly k pieces of sushi and has an associated deliciousness value di.

We are given two positive integers, x and y, and we are asked to find a non-negative integer k such that when we add k to both numbers, their sum equals their bitwise XOR. In other words, we want (x + k) + (y + k) = (x + k) ⊕ (y + k). If no such k exists, we should return -1.

We are given an array of non-negative integers. The only allowed move takes a contiguous segment, replaces it with a single number equal to the MEX of that segment, and shortens the array.

We are given a string of lowercase letters and a limit on the number of swaps we can perform. The goal is to make the string “universal,” meaning that it is lexicographically smaller than its reversal.

Consider a closed broken line with an odd number of vertices $A_1A_2\ldots A_m$.

A zebra-like number is a positive integer whose binary representation looks like 1, 101, 10101, 1010101, ... In other words, the bits alternate and the least significant bit is 1. Let $$z1=1,quad z2=5,quad z3=21,quad z4=85,dots$$ These are exactly the zebra-like numbers.

We are interacting with a process that reveals a binary string one character at a time. After each reveal, we append the new character to a working string, and we are allowed to freely swap any two positions in this working string.

We are asked to count how many strings can be built from given letter counts such that equal letters always occupy positions of the same parity.

We are given a permutation of integers from 1 to n, which means every number in that range appears exactly once in an array of size n. We then perform a sequence of queries, where in each query we replace a specific element of the array with zero.

We are given an array a of length n and a positive integer k. From a, we construct a larger array b by repeating a exactly k times. That is, b has length n k, and its first n elements are the same as a, and the remaining elements repeat a in order.

We are asked to determine how many kilograms of cloudberries are needed to produce a certain number of jars of jam, given that the jam is made by combining equal amounts of berries and sugar, and that during cooking, 25% of the total mass evaporates.

We are given a directed acyclic graph on $n$ vertices with some existing directed edges. The task is to add as few new directed edges as possible so that, after adding them, we can split all edges of the resulting directed multigraph into several Hamiltonian cycles, where each…

Our systems have detected unusual activity coming from your system.

We are dealing with a variant of Nim, a classical two-player impartial game. In standard Nim, the state of the game is defined by piles of stones. On their turn, a player selects any non-empty pile and removes one or more stones. The last player able to move wins.

I’m sorry, but I can’t reliably produce a complete editorial and accepted solution for this problem from the information available here.

We are given a sequence of daily competitions. On each day Monocarp either participates in the competition or spends the day training.

We are given a sorted list of strings, called handles, with the total number $n$ and a target handle $h$. The goal is to determine the 1-based position of $h$ in this list.

We are given a sequence of arrows arranged in a line, each pointing either left or right. Each arrow also has an associated integer reward for painting it red, which can be positive, negative, or zero. Initially, all arrows are blue.

We are asked to generate a password that satisfies three constraints simultaneously: it must contain a specific number of digits, uppercase letters, and lowercase letters, and no two adjacent characters can be the same.

We are given a string made of three possible characters, each position representing a coin of a certain type. For any subsegment of this string, two players play a deterministic picking game where each move consists of choosing a type and then taking all remaining coins of…

We are asked to organize a showmatch for a competitive game involving $2n$ distinct players, each with a unique rating. The key requirement is that every player should be paired with another player whose rating is closest to theirs.

The problem considers a network of $n$ vertices connected by $m$ undirected edges with positive integer weights.

The problem describes a rooted tree with nodes labeled from $1$ to $n$, where node $1$ is the root. Liki and Sasami perform explorations over several days, and each exploration destroys the chosen node and its entire subtree.

We are given a binary string of length 2n+1. The goal is to insert exactly n conditional operators ?: between the characters to form a valid C++-style expression, optionally adding parentheses, and determine if the final expression can evaluate to 1.

We are given a game where n members take turns trying to open l locks using a set of l + k keys. Among these keys, exactly l are genuine and each opens exactly one lock, while the remaining k keys are fake.

In this problem, we are asked to simulate a sequential game where a group of n participants try to unlock l locks using exactly l valid keys. Each participant takes one turn at a time in a fixed cyclic order.

Working

We are given two arrays of equal length. You can think of each index as a station that holds two piles of units: one pile from array a and one pile from array b. The process runs in synchronized rounds, and each round has a fixed structure.

We are asked to construct a permutation of integers from 1 to $n$ with a special property. For each prefix of the permutation, we compute the average of the first $i$ elements, take its ceiling, and call that $ci$.

We are asked to determine who will dig up a buried treasure first between two people, Little B and Little K. The treasure is buried at a fractional depth a.5 meters, which is key: it means the threshold is always .5 meters beyond the integer a.

Let $P(x)$ and $Q(x)$ be polynomials of degrees $m$ and $n$ respectively, with leading coefficients equal to $1$.

Gleb needs to travel along a one-dimensional river segment from position 0 to position $s$ using a kayak. He starts with a given power $k$, and each paddle stroke moves him exactly his current power in the direction he is facing.

We are given a rectangular grid representing a vertical slice of a mountain. Each cell is either empty or contains a hold that Igor can use. The rows are horizontal layers of the mountain, with row 1 at the top and row n at the bottom. The columns are vertical segments.

We are asked to count pairs of integers $(a, b)$ where $1 le a < b le n$ such that the ratio $F(a, b) = frac{text{lcm}(a, b)}{gcd(a, b)}$ is a prime number.

We are given a rectangular hall with n rows and m spots per row where participants can sit. A total of k participants need seats. Desks that are consecutive in the same row form a bench, and the bench's length is the number of consecutive desks.

We are asked to figure out the earliest step at which a specific date, March 1, 2025, can be formed from a sequence of digits drawn one by one.

We are asked to construct a permutation of numbers from 1 to $n$ with a very specific property: for every cyclic shift of the permutation, there should be exactly one fixed point. A fixed point in a permutation is an index $i$ such that the element at that index equals $i$.

We are given a group of students, each with a fixed skill value, and we must split them into teams so that every student belongs to exactly one team. For any team, its strength is defined as the number of students in the team multiplied by the smallest skill inside that team.

We are given a binary string that represents a pie, and we need to split it into contiguous segments such that each segment has the same "beauty." The beauty of a string is defined as the number of places where consecutive characters differ.

The board is an enormous $n times m$ grid where almost every cell is initially uncolored (“green”), except for $k$ cells that are already fixed as either black or white. We are allowed to assign a color to every remaining green cell.

We are given a string representing a drink, made of the letters L, I, and T, each corresponding to a component. The drink is "balanced" when the number of each character is equal. Edmond can insert new letters between two consecutive, different characters.

Represent the tournament by a directed complete graph.

Assume, seeking a contradiction, that

We are given a collection of towers, each with a positive integer height, representing gifts from Asuna's admirers. Asuna evaluates the beauty of her gifts as the height of the tallest tower.

We are given two bit strings of equal length, a and b, representing a password. Lady Bug wants to transform the first string a so that it contains only zeros.

We are given a list of sheep, each with a distinct beauty level. Kamilka can feed every sheep the same number of extra grass bunches, effectively increasing all beauty levels by the same integer $d ge 0$.

Working

We are asked to find the shortest subarray of a given array a whose "beauty" reaches or exceeds a threshold k. The beauty of an array is the maximum XOR value between any pair of its elements.

We are given an array and asked to cut it into exactly k contiguous pieces that together cover the whole array. Once the array is split, each piece has a MEX value, and we care about the worst piece, meaning the smallest MEX among all k segments.

We are filling a $2^n times 2^n$ grid with integers from $1$ to $2^{2n}$, but the order is not row-major or column-major. Instead, the grid is constructed recursively.

We are given a decimal number as a string. We may delete any subset of its digits while preserving the relative order of the remaining digits. The resulting sequence of digits must represent a positive number, although leading zeros are allowed.

We are given a base integer x and a repetition count k. From x, we construct a new number y by writing the decimal representation of x consecutively k times without inserting separators. For example, if x = 52 and k = 3, then y = 525252. If x = 6 and k = 7, then y = 6666666.

We are given a number $k$. The task is to decide whether it is possible to represent every integer $n ge k$ as the sum of a palindromic array of length exactly $k$, where all elements are positive integers.

We are asked to maintain an array under three types of operations: appending an element to the end, reversing the array, and performing a cyclic shift that moves the last element to the front.

We are asked to fill an $n times m$ grid with integers from $1$ to $k$ in such a way that each number occurs exactly the same number of times and no two adjacent cells contain the same number.

We are given an array a of integers and a hidden function f(k, a, l, r) defined as follows: for each index i from l to r, repeatedly divide k by a[i] as long as it is divisible, then add the resulting k to a running total.

Work over the vector space $\mathbb F_2^{mn}$ of all $0$-$1$ configurations on the $m\times n$ rectangle.

We are given a sequence of integers, and for each element in the sequence, we can compute the sum of XORs between it and every other element. Our task is to find which element maximizes this sum. More concretely, for a sequence a = [a1, a2, ...

We are given a recorded sequence of intended drum hits, where each character is either a left hit or a right hit. Each intended hit does not produce a fixed sound length.

We are given an $n times n$ grid that was generated from an unknown permutation $p$ of length $2n$. The construction rule is simple: the value in cell $(i,j)$ is $p{i+j}$.

Normalize the width of the bottom rectangle to $1$, and let the common height of all rectangles be fixed.

We are asked to track the spread of an infection along an infinite line of houses. The infection starts at house 0 on day 0, and each day it spreads to exactly one additional house that is adjacent to any currently infected house.

Each test case describes a “country name” that is stored in an older format consisting of three separate words.

The trees are vertical cylinders.

The task looks almost intentionally trivial: we are given a single integer $x$, which represents a line number in a fixed text image. The problem is essentially asking us to “look” at line $x$ in a predefined visual and output the word printed there.

Before I start drafting the full editorial, I need to clarify one point. Could you briefly describe the mechanics of "Premiere at a Wrong Time"? Specifically, what are we given and what are we supposed to compute or output?

This problem comes from the April Fools Day Contest. The key trick is that the symbol shown as ⊕ is not actually bitwise XOR. The link attached to the symbol leads to a puzzle whose solution reveals that ⊕ should be interpreted as multiplication.

The problem presents an alien numeral system used by the Balikons, where each word corresponds to a number. We are asked to take two numbers expressed in Balikon script, interpret them as standard integers, compute their sum, and then output the result in Balikon script.

We are given a set of points on a plane, each point representing a slime. Megumin can choose a single circle anywhere on the plane and eliminate every slime lying inside it or on its boundary.

Before I start, I need a clarification: the “Problem” section you provided appears garbled and incomplete - there’s a string of symbols and no clear description of what the problem actually is.

The problem is an interactive simulation of a simple Plinko game, where a ball drops through a triangular pegboard and eventually lands in one of ten numbered slots at the bottom. Each round of the game is labeled from Game 1 to Game 10.

I can do that. Before I start, can you confirm: do you want me to reconstruct the full editorial for Codeforces 2095D - Where Am I?, even though the problem statement image and input/output examples are missing?

The problem is unusual because it is more about recognizing a fixed property than performing traditional computation. We are asked to determine how many tests the problem itself has. The input is essentially a prompt asking "how many tests does this problem have?

I can’t reliably reconstruct the full problem from the screenshot link alone, and I don’t want to risk inventing details and giving you a wrong editorial. Please paste the full problem statement (or at least the input/output format and constraints).

The required condition is

We are given n intervals, each defined by a lower and upper bound [li, ri]. From each interval, we can pick a number ai within its bounds. Our goal is to consider all sequences (a1, a2, ...

We are asked to play an interactive guessing game with a student, Alice. She secretly selects a number between 1 and $n$. We do not know her number in advance. Our task is to determine her number by submitting a fixed sequence of queries and then analyzing the responses.